# American Institute of Mathematical Sciences

December  2020, 9(4): 995-1008. doi: 10.3934/eect.2020046

## Stability analysis and optimal control of a stationary Stokes hemivariational inequality

 1 College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3 Department of Mathematics, University of Iowa, Iowa City, IA 52242-1410, USA

* Corresponding author: Weimin Han

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  December 2020 Early access  March 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (No. 11771350), Basic and Advanced Research Project of CQ CSTC (Nos. cstc2016jcyjA0163 and cstc2018jcyjAX0605)

In this paper, we provide stability analysis for a stationary Stokes hemivariational inequality where along the tangential direction of the slip boundary, an inclusion relation involving the generalized subdifferential of a superpotential is specified. With viscous incompressible fluid flows as application background, stability is analyzed for solutions with respect to perturbations in the superpotential and the density of external forces. We also present a result on the existence of a solution to an optimal control problem for the stationary Stokes hemivariational inequality.

Citation: Changjie Fang, Weimin Han. Stability analysis and optimal control of a stationary Stokes hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 995-1008. doi: 10.3934/eect.2020046
##### References:
 [1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [2] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984. [3] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.  doi: 10.5802/aif.280. [4] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. [7] C. Fang, K. Czuprynski, W. Han, X.-L. Cheng and X. Dai, Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition, IMA Journal of Numerical Analysis, (2019). doi: 10.1093/imanum/drz032. [8] C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete Contin. Dyn. Syst., 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036. [9] G. Fichera, Problemi elastostatici con vincoli unilaterali. II. Problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia, 7 (1963/64), 91-140. [10] A. Friedman, Optimal control for variational inequalities, SIAM J. Control Optim., 24 (1986), 439-451.  doi: 10.1137/0324025. [11] H. Fujita, Flow Problems with Unilateral Boundary Conditions, College de France, Lecons, 1993. [12] H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kôkyûroku, 888 (1994), 199–216. [13] H. Fujita and H. Kawarada, Variational inequalities for the Stokes equation with boundary conditions of friction type, Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 11 (1998), 15-33. [14] H. Fujita, H. Kawarada and A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, Lecture Notes Numer. Appl. Anal., Kinokuniya, Tokyo, 14 (1995), 17-31. [15] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-662-12613-4. [16] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. [17] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248. [18] W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023. [19] W. Han and Y. Li, Stability analysis of stationary variational and hemivariational inequalities with applications, Nonlinear Anal. Real World Appl., 50 (2019), 171-191.  doi: 10.1016/j.nonrwa.2019.04.009. [20] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485. [21] J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5. [22] J. Haslinger and P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal., 24 (1995), 105-119.  doi: 10.1016/0362-546X(93)E0022-U. [23] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1. [24] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, Journal of Computational Physics, 128 (1996), 319-330.  doi: 10.1006/jcph.1996.0213. [25] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845. [26] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics, 88. Academic Press, Inc., New York-London, 1980. [27] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971. [28] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302. [29] S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - Supplement, (2013), 545–554. doi: 10.3934/proc.2013.2013.545. [30] S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300.  doi: 10.1023/A:1026555014562. [31] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [32] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995. [33] P. D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410. [34] P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, SpringerVerlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [35] F. Saidi, Non-Newtonian Stokes flow with frictional boundary conditions, Math. Model. Anal., 12 (2007), 483-495.  doi: 10.3846/1392-6292.2007.12.483-495. [36] A. Signorini, Sopra a une questioni di elastostatica, Attidella Società Italiana per il Progresso delle Scienze, (1933). [37] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018. [38] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [39] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564. [40] F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. [41] Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.

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##### References:
 [1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. [2] V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984. [3] H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.  doi: 10.5802/aif.280. [4] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [5] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. [7] C. Fang, K. Czuprynski, W. Han, X.-L. Cheng and X. Dai, Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition, IMA Journal of Numerical Analysis, (2019). doi: 10.1093/imanum/drz032. [8] C. Fang and W. Han, Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow, Discrete Contin. Dyn. Syst., 36 (2016), 5369-5386.  doi: 10.3934/dcds.2016036. [9] G. Fichera, Problemi elastostatici con vincoli unilaterali. II. Problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia, 7 (1963/64), 91-140. [10] A. Friedman, Optimal control for variational inequalities, SIAM J. Control Optim., 24 (1986), 439-451.  doi: 10.1137/0324025. [11] H. Fujita, Flow Problems with Unilateral Boundary Conditions, College de France, Lecons, 1993. [12] H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kôkyûroku, 888 (1994), 199–216. [13] H. Fujita and H. Kawarada, Variational inequalities for the Stokes equation with boundary conditions of friction type, Recent Developments in Domain Decomposition Methods and Flow Problems, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 11 (1998), 15-33. [14] H. Fujita, H. Kawarada and A. Sasamoto, Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, Lecture Notes Numer. Appl. Anal., Kinokuniya, Tokyo, 14 (1995), 17-31. [15] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-662-12613-4. [16] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. [17] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248. [18] W. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica, 28 (2019), 175-286.  doi: 10.1017/S0962492919000023. [19] W. Han and Y. Li, Stability analysis of stationary variational and hemivariational inequalities with applications, Nonlinear Anal. Real World Appl., 50 (2019), 171-191.  doi: 10.1016/j.nonrwa.2019.04.009. [20] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485. [21] J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5. [22] J. Haslinger and P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal., 24 (1995), 105-119.  doi: 10.1016/0362-546X(93)E0022-U. [23] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1. [24] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, Journal of Computational Physics, 128 (1996), 319-330.  doi: 10.1006/jcph.1996.0213. [25] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845. [26] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics, 88. Academic Press, Inc., New York-London, 1980. [27] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971. [28] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302. [29] S. Migórski, A note on optimal control problem for a hemivariational inequality modeling fluid flow, Discrete and Continuous Dynam. Systems - Supplement, (2013), 545–554. doi: 10.3934/proc.2013.2013.545. [30] S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim., 17 (2000), 285-300.  doi: 10.1023/A:1026555014562. [31] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26. Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [32] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995. [33] P. D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410. [34] P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, SpringerVerlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [35] F. Saidi, Non-Newtonian Stokes flow with frictional boundary conditions, Math. Model. Anal., 12 (2007), 483-495.  doi: 10.3846/1392-6292.2007.12.483-495. [36] A. Signorini, Sopra a une questioni di elastostatica, Attidella Società Italiana per il Progresso delle Scienze, (1933). [37] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018. [38] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [39] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564. [40] F. Tröltzsch, Optimal Control of Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. [41] Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.
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